Problem 14 - Elementary and Intermediate Algebra

Question

\(\operatorname{can} 5 x\left(\frac{2}{x}+\frac{4}{5}\right)\) be written as \(5 x \cdot \frac{2}{x}+\frac{4}{5} ?\) Explain.

Step-by-Step Solution

Answer

No, it cannot be written as such; they are not equivalent.

Step 1: Understand the Expression

The given expression is \( 5x \left( \frac{2}{x} + \frac{4}{5} \right) \). We need to determine if this can be written equivalently as \( 5x \cdot \frac{2}{x} + \frac{4}{5} \).

Step 2: Apply the Distributive Property

Apply the distributive property to the initial expression: \( 5x \left( \frac{2}{x} + \frac{4}{5} \right) = 5x \cdot \frac{2}{x} + 5x \cdot \frac{4}{5} \).

Step 3: Simplify Each Term

The first term is \( 5x \cdot \frac{2}{x} = 5 \cdot \frac{2}{1} = 10 \). Since \( x \) in the numerator and denominator cancels out.For the second term: \( 5x \cdot \frac{4}{5} = \frac{5x \cdot 4}{5} = 4x \).

Step 4: Compare and Conclude

The simplified form from Step 3 is \( 10 + 4x \), which does not match the expression \( 5x \cdot \frac{2}{x} + \frac{4}{5} \), which simplifies to \( 10 + \frac{4}{5} \). Therefore, the expressions are not equivalent.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us manage expressions involving products and sums. It essentially tells us how to deal with multiplication across a sum or a difference inside parentheses. In simpler terms, when you have a number or variable outside a set of parentheses, you can distribute, or "spread out," the multiplication to each term inside the parentheses separately.

So, if you have something like \( a(b + c) \), you can write it as \( ab + ac \) by applying the distributive property. This is convenient for simplifying expressions and helps in solving equations.

In our exercise, the expression \( 5x \left( \frac{2}{x} + \frac{4}{5} \right) \) shows this in action, where we multiply \( 5x \) with each term inside the parenthesis separately. This gives us two distinct terms: \( 5x \cdot \frac{2}{x} \) and \( 5x \cdot \frac{4}{5} \). By distributing this way, we prepare the expression for further simplification.

Simplification

Simplification is all about making an expression easier to read and work with. After distributing, it's often necessary to simplify each term. This might involve combining like terms, cancelling terms, or performing any straightforward operations that apply.

Let's take a look at our example after using the distributive property:

• For \( 5x \cdot \frac{2}{x} \), the \( x \) cancels out in the numerator and denominator, leaving us with \( 5 \cdot 2 = 10 \).
• For \( 5x \cdot \frac{4}{5} \), we simplify by dividing, producing \( 4x \).

The result from these simplifications is \( 10 + 4x \), which is a simpler form of the distributed expression and easier to interpret than the initial parentheses-heavy form.

Simplification helps to see the structure and equivalence (or non-equivalence) between expressions, which is vital for problem-solving and verifying solutions.

Equivalence of Expressions

When we talk about equivalence in expressions, we're trying to determine if two different-looking expressions actually mean the same thing or produce the same result under all conditions. This is crucial in algebra, as different forms of an expression can have the same value, or they might not.

With our problem, we started with two expressions: \( 5x \left( \frac{2}{x} + \frac{4}{5} \right) \) and \( 5x \cdot \frac{2}{x} + \frac{4}{5} \). Initially, they might look similar, but through distribution and simplification, we found out they are not equivalent. The first simplified to \( 10 + 4x \), while the second simplified to \( 10 + \frac{4}{5} \).

This example shows why checking for equivalence is important—what starts similarly can end very differently. Equivalence helps in mathematical proofs and validating that the transformations applied to expressions don't change their value.